Suppose you enter 11111000 into a binary calculator. How would it know
whether that is to be 248 or -8? How does the computer begin the division
process? What does it do when it cannot right shift the bits?

The fraction 355/113 is used as an approximation for pi. A question
about converting this fraction to decimal form leads to an interesting
discussion of how to calculate long repeating decimals without having
to divide them all the way out one number at a time.

Twenty persons want to buy a $10 ticket each. Ten of them have a $10
note and others have a $20 note. The person at the ticket counter has
no money to start with. What is the probability that the person at the
ticket counter will not have a change problem?

Is it possible to find a formula that takes X (a large number) and
creates another expression which equals X but is shorter in length
than X? For example, take 390,625 which uses six characters. I need
a formula which calculates something like 5^8, which equals 390,625
but is only three characters instead of six.